Entropy, Free Energy, and Chemical Equilibrium – Chapter 18 Study Notes

Entropy, Free Energy, and Equilibrium

Overview

This chapter explores the thermodynamic concepts of entropy, free energy, and equilibrium, which are central to understanding the spontaneity of chemical processes. The topics include spontaneous processes, entropy (qualitative and quantitative), entropy changes in systems and the universe, predicting spontaneity, and the relationship between free energy and chemical equilibrium.

18.1 Spontaneous Processes

Definition and Examples

A spontaneous process is one that occurs under a specific set of conditions without outside intervention. In contrast, a nonspontaneous process does not occur under those conditions.

• Spontaneous process: Occurs naturally (e.g., water flowing downhill, ice melting above 0°C).

• Nonspontaneous process: Requires energy input (e.g., water freezing at room temperature, a ball rolling uphill).

Comparison Table: Familiar Spontaneous and Nonspontaneous Processes

Spontaneous	Nonspontaneous
Water flowing downhill	Water freezing at room temperature
Reacting violently with water to form sodium hydroxide and hydrogen gas	Sodium hydroxide reacting with hydrogen gas to produce sodium metal and water 2NaOH(aq) + H2(g) → 2Na(s) + 2H2O(l)
A ball rolling downhill	A ball rolling uphill
Rust formation at room temperature	The conversion of rust back to iron metal at room temperature
Ice melting at 10°C	Ice melting at -10°C

Thermochemistry and Spontaneity

• Many exothermic reactions are spontaneous at room temperature. Example: $2H_2(g) + O_2(g) \rightarrow 2H_2O(l)$, $\Delta H^\circ = -890.4\ \text{kJ/mol}$

• Not all exothermic reactions are spontaneous at all temperatures. Example: $H_2O(l) \rightarrow H_2O(s)$, $\Delta H^\circ = -6.01\ \text{kJ/mol}$ (not spontaneous above 0°C)

• Exothermicity favors spontaneity, but is not the sole determining factor.

18.2 Entropy

Qualitative Description of Entropy

Entropy (S) is a measure of the disorder or randomness of a system, or how spread out the system's energy is. The greater the volume a system occupies, the greater its entropy.

• Processes that increase entropy are favored (positive $\Delta S$).

• Entropy increases with increased molecular motion and dispersal of energy.

Quantitative Definition of Entropy

• Entropy is related to the number of possible arrangements (microstates) of a system.

• Boltzmann equation: $S = k \ln W$ where $k$ is the Boltzmann constant ($1.38 \times 10^{-23}\ \text{J/K}$), and $W$ is the number of microstates.

• The Boltzmann constant is related to the gas constant $R$ by $R = k N_A$, where $N_A$ is Avogadro's number.

Example:

If a system can be arranged in 160 different ways, its entropy is higher than if it can only be arranged in 1 way.

18.3 Entropy Changes in a System

Standard Entropy and Calculations

The standard entropy ($S^\circ$) of a substance is its entropy at 1 atm and 25°C. Entropy values are tabulated for many substances.

• Entropy increases from solid to liquid to gas.

• For substances in the same phase, higher molar mass and more complex structure generally mean higher entropy.

Sample Standard Entropy Values at 25°C

Substance	$S^\circ$ (J/K·mol)
H2(g)	131.0
CO2(g)	213.6
Na(s)	51.05
NH3(g)	193.0
HCl(g)	187.0
Au(s)	47.0

Calculating Entropy Change ($\Delta S$)

• For a reaction: $\Delta S^\circ_{rxn} = \sum S^\circ_{products} - \sum S^\circ_{reactants}$

• Example: $\Delta S^\circ_{rxn}$ for $CaCO_3(s) \rightarrow CaO(s) + CO_2(g)$ $\Delta S^\circ_{rxn} = [S^\circ(CaO) + S^\circ(CO_2)] - S^\circ(CaCO_3)$

18.4 Entropy Changes in the Universe

The Second and Third Laws of Thermodynamics

• Second Law: For any spontaneous process, the total entropy change of the universe ($\Delta S_{univ}$) must be positive. $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} > 0$

• Third Law: The entropy of a perfect crystalline substance at absolute zero (0 K) is zero. $S = k \ln 1 = 0$

Entropy Change for the Surroundings

• $\Delta S_{surr}$ is directly proportional to $-\Delta H_{sys}$ and inversely proportional to temperature:

$\Delta S_{surr} = -\frac{\Delta H_{sys}}{T}$

18.5 Predicting Spontaneity

Gibbs Free Energy ($G$)

The Gibbs free energy is a thermodynamic function that combines enthalpy and entropy to predict spontaneity at constant temperature and pressure.

• $G = H - TS$

• Change in free energy: $\Delta G = \Delta H - T\Delta S$

• If $\Delta G < 0$, the process is spontaneous.

• If $\Delta G > 0$, the process is nonspontaneous.

• If $\Delta G = 0$, the system is at equilibrium.

Temperature Dependence

• Spontaneity can depend on temperature, especially when $\Delta H$ and $\Delta S$ have the same sign.

• For example, if $\Delta H > 0$ and $\Delta S > 0$, the process is spontaneous only at high temperatures.

Example Calculation

For a reaction with $\Delta H = 199.5\ \text{kJ/mol}$ and $\Delta S = 0.476\ \text{kJ/K·mol}$, the temperature above which the reaction is spontaneous is:

$T = \frac{\Delta H}{\Delta S} = \frac{199.5}{0.476} = 419\ \text{K}$

18.6 Free Energy and Chemical Equilibrium

Relationship Between $\Delta G$ and Equilibrium Constant ($K$)

• At equilibrium: $\Delta G = 0$

• Relationship: $\Delta G = \Delta G^\circ + RT \ln Q$

• At equilibrium ($Q = K$): $\Delta G^\circ = -RT \ln K$

• If $\Delta G^\circ < 0$, products are favored ($K > 1$).

• If $\Delta G^\circ > 0$, reactants are favored ($K < 1$).

Example Calculation

For the reaction $H_2O(l) \rightarrow 2H_2(g) + O_2(g)$ with $\Delta G^\circ = 474.4\ \text{kJ/mol}$ at 25°C:

$\Delta G^\circ = -RT \ln K$ $474.4 = -8.314 \times 298 \ln K$ $\ln K = -191.5$ $K = e^{-191.5} \approx 7 \times 10^{-84}$

18.7 Thermodynamics in Living Systems

Application in Biological Systems

Thermodynamic principles, including entropy and free energy, are essential for understanding biochemical processes such as metabolism, energy transfer, and equilibrium in living organisms.

• Living systems maintain order and function by coupling nonspontaneous processes with spontaneous ones.

• ATP hydrolysis is a key example of a spontaneous process that drives cellular work.

Additional info: These notes are based on textbook slides for Chapter 18 of "Chemistry" by Julia Burdge, Sixth Edition, and cover all major learning objectives for a General Chemistry course on thermodynamics and equilibrium.